The [[4,2,2]] quantum error detection code is a simple quantum error detection code, which needs 4 physical qubits for 2 logical qubits and has a code distance 2. It is the smallest stabilizer code that can detect X- and Z-errors [1]}. One possible set of code words for the [[4,2,2]] error detection code is:
\(\mathcal {E}_{\rm [[4,2,2]]}=\operatorname{span}\left\lbrace \begin{array}{l}|00\rangle _{L}=\frac{1}{\sqrt{2}}(|0000\rangle +|1111\rangle ) \\|01\rangle _{L}=\frac{1}{\sqrt{2}}(|0110\rangle +|1001\rangle ) \\|10\rangle _{L}=\frac{1}{\sqrt{2}}(|1010\rangle +|0101\rangle ) \\|11\rangle _{L}=\frac{1}{\sqrt{2}}(|1100\rangle +|0011\rangle )\end{array}\right\rbrace \) 
